316 XV NORMED ALGEBRAS AND SPECTRAL THEORY An element x of A is of the form expO) for some y e A if and only if x is contained in a connected commutative subgroup of G. (To prove that this condition is sufficient, reduce to the case where A is commutative.) 10. Let A be a Banach algebra with unit element e. Let G be the group of invertible elements of A, and G0 the connected component of the neutral element of G. (a) If x e A is such that x" = e for some integer n > 0, show that x E G0. (Observe that the spectrum of x is finite, and deduce that the set of complex numbers A such that Xx + (1 — X)e is invertible is connected.) (b) Suppose that A is commutative. Show that every element of G/G0 (other than the neutral element) has infinite order. (Remark that if x e G and x" e G0, then x" = exp(j) for some ye A, whence (x Qxp(—(l/n)y))n = e; now use (a).) 11. Let A be a commutative Banach algebra with unit element e. Let x e A and let K = SpAC*:). Suppose that K has the following property: there exists a fundamental system of neighborhoods Un of K such that the frontier of each Un is the union of a certain number of disjoint simple closed curves which are images of circuits ynk (1 ^ k ^Pn); the frontier of Un+1 is contained in Un, and if a function F is analytic in a neighborhood of On — Un+i, with values in a complex Banach space, then finally, if F is analytic in Un, then £ f F(f) dt, - 0, and for all z e Un, *=1 Jynk F(z) = — V f 27TI jft'l JVn.. . (These properties can be proved by using Problem 4(b) of the Appendix to Chapter IX; see Chapter XXIV for another proof.) (a) Let V be a neighborhood of K, and/an analytic function on V with values in C. Show that, for each n such that Un c V, the element — £ f 2m * = i V of A is independent of n. We denote it byf(x). If W is another neighborhood of K, and g an analytic function in W with values in C, and if /(£) = g(£) for all £ in some neighborhood of K contained in V n W, then f(x) = g(x). (b) If /(£) - 1, then /(*) = e. If /(£) = £, then f(x) == x. (Use Cauchy's theorem (9.6.3) to reduce the problem to integration around a circle with center 0 and radius >p(x), so as to be able to expand (£e — x)'1 as a power series in £~1.) (c) If h =/-f g (resp. h —fg}> where/and g are analytic in a neighborhood of K, then h(x) = f(x)-+• g(x) (resp. h(x)=f(x)g(x)). (For the second assertion, express f(x) in terms of a neighborhood Un and g(x) in terms of UnH. j; observe that -i _ l and use the Lebesgue-Fubini theorem.) (d) If/is analytic in a neighborhood of K, with values in C, then Sp(/(#)) = /(Sp(x)). (Observe that if A e Sp(x) we have/(A)e -f(x) = (Xe - x)g(x) for a suitably chosen function g.) In particular, f(x) is invertible in A if and only if /(£) ^ 0 on Sp(#).n particular, the image of C under the